Universal property of direct product

Question

I have been studying category theory and I saw that it is written that in universal property of direct product(for modules),direct product is a final object. I know that an object Y in a category is called final if there exists exactly one morphism from X to Y for all objects X in the category. Now i can't understand in the universal property of direct product,what is the category,objects of category and morphisms to imply the given result.Please explain

Answer 1

Consider a category $\mathcal{C}$ with binary products and two objects $A,B$ of $\mathcal{C}$. Let us define a category $\mathcal{C}_{A,B}$ as follows:

• objects: the objects of $\mathcal{C}_{A,B}$ are diagrams of the form

$$A\leftarrow P\rightarrow B$$

• morphism: a morphism from a diagram $$A\xleftarrow{f} Q\xrightarrow{g} B$$ to a diagram $$A\xleftarrow{f'} P\xrightarrow{g'} B$$ is a morphism of $\mathcal{C}$, call it $\sigma$, from $Q$ to $P$ such that $$f'\circ\sigma=f\qquad\text{and }\qquad g'\circ\sigma=g$$

• composites and identities: the same composites and identities as in $\mathcal{C}$

Then the universal property of products says that, for every product-diagram $(Q, f',g')$ there exists a unique factorization $\sigma$ through the product $(P, f,g)$ of $A$ and $B$, which means, for every object $(Q, f', g')$ of $\mathcal{C}_{A,B}$, there exists a unique morphism having the product of $A$ and $B$ as target, hence the product is terminal in the category $\mathcal{C}_{A,B}$